Question
Answer
$$\dfrac{\sqrt{10}-\sqrt{5}}{\sqrt{10}+\sqrt{5}}=3-2\sqrt{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{10}-\sqrt{5}}{\sqrt{10}+\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{10}-\sqrt{5}}{\sqrt{10}+\sqrt{5}}\frac{\sqrt{10}-\sqrt{5}}{\sqrt{10}-\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10-5\sqrt{2}-5\sqrt{2}+5}{10-5\sqrt{2}+5\sqrt{2}-5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15-10\sqrt{2}}{5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{3-2\sqrt{2}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}3-2\sqrt{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{10}- \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{10}- \sqrt{5}\right) } \cdot \left( \sqrt{10}- \sqrt{5}\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot- \sqrt{5}\color{blue}{- \sqrt{5}} \cdot \sqrt{10}\color{blue}{- \sqrt{5}} \cdot- \sqrt{5} = \\ = 10- 5 \sqrt{2}- 5 \sqrt{2} + 5 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{10} + \sqrt{5}\right) } \cdot \left( \sqrt{10}- \sqrt{5}\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot- \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot \sqrt{10}+\color{blue}{ \sqrt{5}} \cdot- \sqrt{5} = \\ = 10- 5 \sqrt{2} + 5 \sqrt{2}-5 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 5. |
| ⑤ | Remove 1 from denominator. |
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